Mix together ½ cup softened butter, ¼ cup vegetable oil, ½ cup white sugar, and 2 tablespoons powdered sugar for 3-4 minutes until creamy.
Add 1 egg, ½ teaspoon vanilla extract, and ¼ teaspoon almond extract, and mix again for 2-3 minutes.
In a separate bowl, mix 2 ⅓ cups all-purpose flour, ¼ teaspoon cream of tartar, ¼ teaspoon salt, and 1 teaspoon baking powder.
Add the dry ingredients to the wet ingredients and mix until the flour disappears, being careful not to overmix.
Roll the dough into balls of about ¼ cup each, slightly flatten them but keep them at least ½ inch thick.
Bake the cookies at 350°F for 9-10 minutes. Let them continue to cool on the baking sheet for an additional 20 minutes.
For the frosting, mix ½ cup softened butter for 2-3 minutes. Add 2 cups powdered sugar, ¼ teaspoon almond extract (optional), and 1 teaspoon vanilla extract. Add up to 2 tablespoons of heavy whipping cream or milk to thin the frosting if needed.
Pipe the frosting onto the cooled cookies.
